ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

NAVIER-STOKES EQUATIONS IN THE HALF-SPACE IN VARIABLE EXPONENT SPACES OF CLIFFORD-VALUED

FUNCTIONS

RUI NIU, HONGTAO ZHENG, BINLIN ZHANG Communicated by Vicentiu Radulescu

Abstract. In this article, we study the steady generalized Navier-Stokes equations in a half-space in the setting of variable exponent spaces. We first establish variable exponent spaces of Clifford-valued functions in a half-space.

Then, using this operator theory together with the contraction mapping prin- ciple, we obtain the existence and uniqueness of solutions to the stationary Navier-Stokes equations and Navier-Stokes equations with heat conduction in a half-space under suitable hypotheses.

1. Introduction

Since Kov´aˇcik and R´akosn´ık [24] first studied the spacesL^{p(x)}andW^{k,p(x)}, more
and more attention are paid to Lebesgue and Sobolev variable exponent spaces and
their applications to differential equations. See [7, 8] for basic properties of variable
exponent spaces and [21, 33] for recent overviews of differential equations with
variable growth. It is well-known that one of the reasons that forced the rapid
expansion of the theory of variable exponent function spaces has been the models
of electrorheological fluids introduced by Rajagopal and R˚uˇziˇcka [29, 30], which
can be described by the boundary-value problem for the generalized Navier-Stokes
equations. In the setting of variable exponent spaces, Diening et al. [5] proved
the existence and uniqueness of strong and weak solutions of the Stokes system
and Poisson equations for bounded domains, the whole-space and the half-space,
respectively.

In the previous decades, the study of these spaces has been stimulated by prob- lems in elastic mechanics, calculus of variations and differential equations with variable growth conditions, see [9, 12, 31, 32, 34, 35, 36] and references therein.

As a powerful tool for solving elliptic boundary value problems in the plane, the methods of complex function theory play an important role. One way to extend these ideas to higher dimension is to begin with a generalization of algebraic and geometrical properties of the complex numbers. In this way, Hamilton studied the algebra of quaternion in 1843. Further generalizations were introduced by Clifford

2010Mathematics Subject Classification. 30G35, 35J60, 35Q30, 46E30, 76D03.

Key words and phrases. Clifford analysis; variable exponent; Navier-Stokes equations;

half-space.

c

2017 Texas State University.

Submitted January 28, 2017. Published April 5, 2017.

1

in 1878. He initiated the so-called geometric algebras or Clifford algebras, which are generalizations of the complex numbers, the quaternions, and the exterior algebras, see [19]. After that, Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. Clifford analysis as an active branch of mathematics concerned with the study of Dirac equation or of a generalized Cauchy-Riemann system, in which solutions are de- fined on domains in the Euclidean space and take values in Clifford algebras, see the monograph of Brackx et al. [1]. It is worthy mentioning that G¨urlebeck and Spr¨oßig [14, 15] developed an operator calculus, which is analogous to the known complex analytic approach in the plane and based on three operators: a Cauchy- Riemann-type operator, a Teodorescu transform, and a generalized Cauchy-type integral operator, to investigate elliptic boundary value problems of fluid dynam- ics over bounded and unbounded domains, especially the Navier-Stokes equations and related equations. Of course, there are a number of unsolved basic problems involving the Navier-Stokes equations. This is mainly due to the problem concern- ing the solvability of the corresponding linear Stokes equations over domains, see [2, 16]. As Galdi [20] pointed out, the study of the Stokes problem in the half-space possesses an independent interest and it will be fundamental for the treatment of other linear and nonlinear problems when the region of flow is either an exterior domain or a domain with a suitable unbounded boundary.

On the one hand, Diening et al. [4] studied the following model introduced in [29, 30] to describe motions of electrorheological fluids:

−divM(Du) + (u· ∇)u+∇π=f x∈Ω divu= 0 x∈Ω

u= 0 x∈∂Ω,

(1.1)

where Ω is a bounded domain with Lipschitz boundary∂Ω, f ∈(W_{0}^{1,p(x)}(Ω))^{∗} =
W^{−1,p}^{0}^{(x)}(Ω), 2n/(n+ 2) < p_{−} ≤ p_{+} <∞ and the operator M satisfies certain
natural variable growth conditions. The authors obtained the existence of weak
solutions in (W_{0}^{1,p(x)}(Ω))^{n}×L^{s}_{0}(Ω), heres:= min

(p_{+})^{0}, np_{−}/2(n−p_{−}) ifp_{−}< n
and s := (p+)^{0} otherwise, L^{s}_{0}(Ω) := {π ∈ L^{s}(Ω) : R

Ωπdx = 0}. Diening et al.

[5] studied the Stokes and Poisson problem in the context of variable exponent spaces in bounded domains and in the whole space. In the half-space case, the authors employed a localization technique to reduce the interior and the boundary regularity to regularity results on the half-space. While it should be pointed out that our attempt is to give a unified approach to deal with physical problems modelled by the generalized Navier–Stokes equations, which is quite different with approaches of some authors, for example, we refer to the monograph [3].

On the other hand, it is natural to focus on theA-Dirac equations if one interests
in extending the classical Dirac equations. In [26, 27], Nolder first introduced the
general nonlinear A-Dirac equations DA(x, Du) = 0 which arise in the study of
many phenomena in physical sciences. Moreover, he developed some tools for the
study of weak solutions to nonlinearA-Dirac equations in the spaceW_{0}^{1,p}(Ω,C`n).

Inspired by his works, Fu and Zhang in [10, 11] were devoted to the the existence of weak solutions for the general nonlinearA-Dirac equations with variable growth.

For this purpose, the authors established a theory of variable exponent spaces of Clifford-valued functions with applications to homogeneous and non-homogeneous

A-Dirac equations, see also [37]. Recently, Fu et al. [9, 28, 38] established a Hodge- type decomposition of variable exponent Lebesgue spaces of Clifford-valued func- tions with applications to the Stokes equations, the Navier-Stokes equations and theA-Dirac equationsDA(Du) = 0. By using the Hodge-type decomposition and variational methods, Molica Bisci et al. [25] studied the properties of weak solutions to the homogeneous and nonhomogeneousA-Dirac equations with variable growth.

Motivated by the above works, we study of Navier-Stokes equations in a half- space in variable exponent spaces of Clifford-valued functions. To the best of our knowledge, this is the first time to investigate Navier-Stokes equations over un- bounded domains in such spaces. To this end, we first establish variable expo- nent spaces of Clifford-valued functions in the half-space. Then, using an iteration method which requires the solution of a Stokes-problem in every step of iteration, we study the existence and uniqueness of Navier-Stokes equations in a half-space.

There is no doubt that we encounter serious difficulties, for instance, the Sobolev embedding is not compact in a half-space, and operator theory in variable exponent spaces of Clifford-valued functions in a half-space is still unknown. Anyway, our attempt would be a meaningful exploration in the study of fluid dynamics, and the whole treatment applies to a much larger class of elliptic problems.

This article is organized as follows. In Section 2, we start with a brief summary of
basic knowledge of Clifford algebras and then investigate basic properties of variable
exponent spaces of Clifford-valued functions in a half-space. In Section 3, with the
help of the results of Diening et al. [5], we prove the existence and uniqueness of
the Stokes equations in the context of variable exponent spaces in a half-space. In
Section 4, we present an iterative method for the solution of the stationary Navier-
Stokes equations. Using the contraction mapping principle, we prove the existence
and uniqueness of solutions to the Navier-Stokes equations in W_{0}^{1,p(x)}(R^{N}+,C`n)×
L^{p(x)}(R^{N}+,R) under certain hypotheses. In Section 5, using the contracting mapping
principle, we obtain the existence and uniqueness of solutions for the Navier-Stokes
problem with heat conduction under some appropriate assumptions.

2. Preliminaries

2.1. Clifford algebras. We first recall some related notions and results concerning Clifford algebras. For a detailed account we refer to [14, 15, 26, 27].

Let C`n be the real universal Clifford algebras overR^{n}. Denote C`n by
C`n= span{e0,e1,e2, . . . ,en, e1e2, . . . ,e_{n−1}en, . . . ,e1e2· · ·en}

where e0 = 1(the identity element inR^{n}), {e1,e2, . . . ,en}is an orthonormal basis
ofR^{n} with the relation eiej+ ejei=−2δije0. Thus the dimension of C`n is 2^{n}. For
I={i1, . . . , ir} ⊂ {1, . . . , n}with 1≤i1< i2<· · ·< in≤n, put eI = ei1ei2· · ·eir,
while forI=∅, e_{∅}= e_{0}. For 0≤r≤nfixed, the space C`^{r}_{n} is defined by

C`^{r}_{n}= span{e_{I} :|I|:= card(I) =r}.

The Clifford algebras C`n is a graded algebra as
C`n=⊕rC`^{r}_{n}.

Any elementa∈C`n may thus be written in a unique way as
a= [a]_{0}+ [a]_{1}+· · ·+ [a]_{n}

where [ ]r : C`n →C`^{r}_{n} denotes the projection of C`n onto C`^{r}_{n}. In particular, by
C`^{2}_{n} =Hwe denote the algebra of real quaternion. It is customary to identifyR
with C`^{0}_{n} and identifyR^{n} with C`^{1}_{n} respectively. This means that each elementx
ofR^{n} may be represented by

x=

n

X

i=1

xiei.

Foru∈C`n, we denotes by [u]0 the scalar part of u, that is the coefficient of the element e0. We define the Clifford conjugation as follows:

ei_{1}ei_{2}· · ·ei_{r} = (−1)^{r(r+1)}^{2} ei_{1}ei_{2}· · ·ei_{r}.
We denote

(A, B) = AB

0.

Then an inner product is thus obtained, giving to the norm| · |on C`_{n} given by

|A|^{2}=
AA

0.

From [15], we know that this norm is submultiplicative: |AB| ≤ C(n)|AkB|,
whereC(n) is a positive constant depending only onnand no more than 2^{n/2}.

In what follows, we let R^{n}+ ={(x1, . . . , xn)∈ R^{n} :xn >0} and Σ = ∂R^{n}+. A
Clifford-valued function u:R^{n}+ →C`n can be written as u= ΣIuIeI, where the
coefficientsuI :R^{n}+→Rare real-valued functions.

The Dirac operator on the Euclidean space used here is introduced by D=

n

X

j=1

ej

∂

∂x_{j} :=

n

X

j=1

ej∂j.

Ifuis a real-valued function defined onR^{n}+, thenDu=∇u= (∂_{1}u, ∂_{2}u, . . . , ∂_{n}u).

Moreover, D^{2} = −∆, where ∆ is the Laplace operator which operates only on
coefficients. A function is left monogenic if it satisfies the equationDu(x) = 0 for
eachx∈R^{n}+. A similar definition can be given for right monogenic function. An
important example of a left monogenic function is the generalized Cauchy kernel

G(x) = 1 ωn

x

|x|^{n},

whereωn denotes the surface area of the unit ball inR^{n}. This function is a funda-
mental solution of the Dirac operator. Basic properties of left monogenic functions
one can refer to [11, 17] and references therein.

2.2. Variable exponent spaces of Clifford-valued functions. Next we recall
some basic properties of variable exponent spaces of Clifford-valued functions. In
what follows, we use the short notation L^{p(x)}(R^{N}+), W^{1,p(x)}(R^{N}+), etc., instead of
L^{p(x)}(R^{N}+,R),W^{1,p(x)}(R^{N}+,R), etc. Throughout this paper we always assume (un-
less declared specially)

p∈ P^{log}(R^{n}+)and1< p_{−}:= inf

x∈R^{n}+

p(x)≤p(x)≤ sup

x∈R^{n}+

p(x) =:p_{+}<∞. (2.1)
where P^{log}(R^{n}+) is the set of exponent psatisfying the so-called log-H¨older conti-
nuity, i.e.,

|p(x)−p(y)| ≤ C

log(e+|x−y|^{−1}), |p(x)−p(∞)| ≤ C
log(e+|x|^{−1})

hold for allx, y∈R^{N}+, wherep(∞) = lim_{|x|→∞}p(x), see [3, 5]. LetP(R^{n}+) be the
set of all Lebesgue measurable functions p: R^{n}+ → (1,∞). Givenp∈ P(R^{n}+) we
define the conjugate functionp^{0}(x)∈ P(R^{n}+) by

p^{0}(x) = p(x)

p(x)−1 for eachx∈R^{N}+.
The variable exponent Lebesgue spaceL^{p(x)}(R^{n}+) is defined by

L^{p(x)}(R^{n}+) =

u∈ P(R^{n}+) :
Z

R^{n}+

|u|^{p(x)}dx <∞ ,

with the norm

kuk_{L}p(x)(R^{n}_{+})= inf
t >0 :

Z

R^{n}_{+}

|u

t|^{p(x)}dx≤1 .

The variable exponent Sobolev spaceW^{1,p(x)}(R^{n}+) in a half-space is defined by
W^{1,p(x)}(R^{n}+) =

u∈L^{p(x)}(R^{n}+) :|∇u| ∈L^{p(x)}(R^{n}+) ,
with the norm

kuk_{W}1,p(x)(R^{n}+)=k∇uk_{L}p(x)(R^{n}+)+kuk_{L}p(x)(R^{n}+). (2.2)
Denote W_{0}^{1,p(x)}(R^{n}+) by the completion of C_{0}^{∞}(R^{n}+) in W^{1,p(x)}(R^{n}+) with respect
to the norm (2.2). The space W^{−1,p(x)}(R^{n}+) is defined as the dual of the space
W_{0}^{1,p}^{0}^{(x)}(R^{n}+). For more details we refer to [3, 7, 8] and reference therein.

In the following, we say that u ∈ L^{p(x)}(R^{n}+,C`n) can be understood coordi-
nate wise. For example, u ∈ L^{p(x)}(R^{n}+,C`n) means that {uI} ⊂ L^{p(x)}(R^{n}+) for
u = ΣIuIeI ∈ C`n with the norm kuk_{L}p(x)(R^{n}_{+},C`_{n}) = P

IkuIk_{L}p(x)(R^{n}_{+}). In the
same way, spacesW^{1,p(x)}(R^{n}+,C`n), W_{0}^{1,p(x)}(R^{n}+,C`n),C_{0}^{∞}(R^{n}+,C`n), etc., can be
understood similarly. In particular, the space L^{2}(R^{n}+,C`_{n}) can be converted into
a right Hilbert C`_{n}-module by defining the following Clifford-valued inner product
(see [14, Definition 3.74])

f, g

C`_{n} =
Z

R^{n}+

f(x)g(x)dx. (2.3)

Remark 2.1. Following the same arguments as in [10, 37], we can calculate easily
thatkuk_{L}p(x)(R^{n}+,C`_{n})is equivalent to the normk|u|k_{L}p(x)(R^{n}+). Furthermore, we also
prove that for everyu∈W_{0}^{1,p(x)}(R^{n}+,C`n),kDuk_{L}p(x)(R^{n}_{+},C`_{n})is an equivalent norm
ofkuk_{W}1,p(x)

0 (R^{n}+,C`_{n}).

Lemma 2.2 ([10]). Assume thatp(x)∈ P(R^{n}+). Then
Z

R^{n}+

|uv|dx≤C(n, p)kuk_{L}p(x)(R^{n}+,C`n)kvk_{L}p0(x)(R^{n}+,C`n)

for everyu∈L^{p(x)}(R^{n}+,C`_{n})andv∈L^{p}^{0}^{(x)}(R^{n}+,C`_{n}).

Lemma 2.3([10, 11]). Ifp(x)∈ P(R^{n}+), thenL^{p(x)}(R^{n}+,C`_{n})andW^{1,p(x)}(R^{n}+,C`_{n})
are reflexive Banach spaces.

Based on the Cauchy kernel G(x) we can introduce the Teodorescu operator.

There exist a number of applications and methods based on the properties of this
Teodorescu operator. But in our case of considering the domain Ω as an unbounded
domain, the main problem in applying this operator is that the Cauchy kernel
does not have good enough behaviour near infinity. For example, the Teodorescu
operator is an unbounded operator over the usual function spaces on Ω. In this
paper, we will follow the idea from [2, 18] by using add-on terms to the Cauchy
kernel. More precisely, we choose a fixed pointz lying in the complement of R^{N}+.
Then we consider the following operators.

Definition 2.4 ([2, 14, 15]). (i) Letu∈C(R^{n}+,C`n). The Teodorescu operator is
defined by

T u(x) = Z

R^{n}_{+}

Kz(x, y)u(y)dy,

where Kz(x, y) = G(x−y)−G(y−z), G(x) is the above-mentioned generalized Cauchy kernel.

(ii) Letu∈C^{1}(R^{n}+,C`_{n})∩C(R^{n}+,C`_{n}). Theboundary operatoris defined by
F u(x) =

Z

Σ

Kz(x, y)α(y)u(y)dSy, whereα(y) denotes the outward normal unit vector aty.

(iii) Letu∈L^{1}_{loc}(R^{n}). Then the Hardy-Littlewood maximal operator is defined
by

M u x

= sup

x∈Q

1

|Q|

Z

Q

|u(y)|dy.

for allx∈R^{n}, where the supremum is taken over all cubes (or ball)Q⊂R^{n} which
containx.

The Teodorescu operator was first introduced in [18] and the operator properties
in the scale ofW^{k,2}-spaces were given in [23], see also [2] for the corresponding op-
erator properties in theW^{k,q}-spaces over unbounded domains. Its main advantage
is a faster decay to infinity of the kernel.

Lemma 2.5. (see [3])Let x∈R^{n},δ >0andu∈L^{1}_{loc}(R^{n}). Then
Z

B(x,δ)

1

|x−y|^{n−1}|u(y)|dy≤C(δ)M u(x).

whereC(δ)>0is a positive constant. Moreover, ifu∈L^{p(x)}(R^{n})withkukp(x)≤1,

then Z

R^{n}\B(x,δ)

1

|x−y|^{n−1}|u(y)|dy≤C(n, p, δ,|B|).

whereC(n, p, δ,|B|)is a positive constant.

Lemma 2.6 ([3]). Let p(x)satisfy (2.1). ThenM is bounded inL^{p(x)}(R^{n}).

Lemma 2.7 ([18]). Let u∈C^{1}(R^{n}+,C`_{n}). Then

∂_{k}T u(x) = 1
ωn

Z

R^{n}_{+}

∂

∂xk

G(x−y)u(y)dy+u(x)
n e_{k}.

Lemma 2.8([3]). LetΦbe a Calder´on-Zygmund operator with Calder´on-Zygmund
kernelK onR^{n}×R^{n}. Then Φis bounded onL^{p(x)}(R^{n}).

Lemma 2.9. The following operators are continuous linear operators:

(i) T :L^{p(x)}(R^{n}+,C`n)→W^{1,p(x)}(R^{n}+,C`n).

(ii) Te:W^{−1,p(x)}(R^{n}+,C`n)→L^{p(x)}(R^{n}+,C`n).

Proof. (i) We divide the proof into two parts:

Step 1: The operator∂_{k}T : L^{p(x)}(R^{n}+,C`_{n})→L^{p(x)}(R^{n}+,C`_{n}) is continuous. By
Lemma 2.7 we have foru∈C_{0}^{∞}(R^{n}+,C`n)

∂kTu(x) = 1
ω_{n}

Z

R^{n}+

∂

∂x_{k}Kz(x, y)u(y)dy+u(x)
n ek.
LetK(x, y) = _{ω}^{1}

n

∂

∂x_{k}Kz(x, y). Since _{ω}^{1}

n

∂

∂x_{k}Kz(x, y) = _{ω}^{1}

n

∂

∂x_{k}G(x−y) and

∂

∂x_{k}G(x−y) = 1

|x−y|^{n}

ek−n

n

X

i=1

(xk−yk)(xi−yi)

|x−y|^{2} ei

,

we obtain

∂

∂xk

G(x−y)

≤ n^{2}+ 1

|x−y|^{n}, (k= 1, . . . , n).

Notice that

Z

S_{1}

ek−n

n

X

i=1

(xk−yk)(xi−yi)

|x−y|^{2} ei

dS= 0,

where S_{1}={y∈R^{n}+:|x−y|= 1}. Then it is easy to verify thatK(x, y) satisfies
the following properties:

(a) |K(x, y)| ≤C|x−y|^{−n};
(b) K t(x, y)

=t^{−n}K(x, y), t >0;

(c) R

S_{1}K(x, y)dS= 0.

Now we defineu(x) = 0 for x∈R^{n}\R^{n}+. Then K(x, y) satisfies the conditions of
Calder´on-Zygmund kernal onR^{n}×R^{n}. By Theorem 2.13, we know the inequality
can be extended toL^{p(x)}(R^{n}+,C`n). Therefore, we obtain by Lemma 2.5 and Lemma
2.6

k 1 ωn

Z

Ω

∂_{k,x}G(x−y)u(y)dyk_{L}p(x)(R^{n}+,C`_{n})≤C(n, p)kuk_{L}p(x)(R^{n}+,C`_{n}) (2.4)
On the other hand,

ku(x)

n ekk_{L}p(x)(R^{n}+,C`_{n})≤ 1

nkuk_{L}p(x)(R^{n}+,C`_{n}) (2.5)
Combining (2.3) with (2.5), we obtain

k∂kTuk_{L}p(x)(R^{n},Cln)≤C(n, p)kuk_{L}p(x)(R^{n}+,C`n).

Step 2: The operator T : L^{p(x)}(R^{n}+,C`n) → L^{p(x)}(R^{n}+,C`n) is continuous. We
defineu(x) = 0 forx∈R^{n}\R^{n}+. Since

|G(x−y) +G(y−z)| ≤ 1
ω_{n}

1

|x−y|^{n−1} + 1

|y−z|^{n−1}

,

we have

|T u(x)| ≤C(n) Z

R^{n}+

(|G(x−y)|+|G(y−z)|)|u(y)|dy

≤CZ

R^{n}+

1

|x−y|^{n−1}|u(y)|dy+
Z

R^{n}+

1

|y−z|^{n−1}|u(y)|dy
.

Then by Lemmas 2.5 and 2.6, we obtain

kT uk_{L}p(x)(R^{n}+,C`n)≤C(n, p)kuk_{L}p(x)(R^{n}+,C`n).
Finally, combining Step 1 with Step 2, we have

kT uk_{W}1,p(x)(R^{n}+,C`n)=kT uk_{L}p(x)(R^{n}+,C`n)+

n

X

k=1

k∂_{k}Tuk_{L}p(x)(R^{n}+,C`n)

≤C(n, p)kuk_{L}p(x)(R^{n}+,C`n).
Then we obtain the desired conclusion (i).

(ii) In view of [3, Proposition 12.3.2], we know that for eachf ∈W^{−1,p(x)}(R^{n}+),
there existsfk∈L^{p(x)}(R^{n}+), k= 0,1, . . . , n, such that

hf, ϕi=

n

X

k=0

Z

R^{n}+

fk

∂ϕ

∂x_{k}dx, (2.6)

for allϕ∈W^{1,p}

0(x)

0 (R^{n}+). Moreover,kfk_{W}−1,p(x)(R^{n}_{+})is equivalent toPn

k=0kfkk_{L}p(x)(R^{n}_{+}).
Obviously, for every f ∈W^{−1,p(x)}(R^{n}+,C`_{n}) the equality (2.5) still holds for f_{k} ∈
L^{p(x)}(R^{n}+,C`n), k = 0,1, . . . , n. Moreover, kfk_{W}−1,p(x)(R^{n}+,C`n) is equivalent to
Pn

k=0kfkk_{L}p(x)(R^{n}_{+},C`_{n}). On the other hand, by [3, Proposition 12.3.4], it is easy
to show that the spaceC_{0}^{∞}(R^{n}+,C`n) is dense inW^{−1,p(x)}(R^{n}+,C`n). Thus we may
choose

u^{j} =u^{j}_{0}+

n

X

k=1

∂u^{j}_{k}

∂xk

,

where u^{j}_{0}, u^{j}_{k} ∈ C_{0}^{∞}(R^{n}+,C`_{n}), such that ku^{j}−fk_{W}−1,p(x)(R^{n}+,C`_{n}) → 0 andku^{j}_{k}−
f_{k}k_{L}p(x)(R^{n}+,C`n)→0 as j→ ∞, wherek= 0,1, . . . , n. Here, we are using the fact
thatC_{0}^{∞}(R^{n}+,C`n) is dense inL^{p(x)}(R^{n}+,C`n)(see [3]). Then we consider

T u^{j}=
Z

R^{n}+

Kz(x, y)u^{j}(y)dy.

Then we have
T u^{j} =

Z

R^{n}+

K_{z}(x, y)
u^{j}_{0}(y) +

n

X

k=1

∂

∂yk

u^{j}_{k}(y)
dy

= Z

R^{n}_{+}

K_{z}(x, y)u^{j}_{0}(y)dy−

n

X

k=1

Z

R^{n}_{+}

∂

∂yk

K_{z}(x, y)u^{j}_{k}(y)dy.

Since Z

R^{n}_{+}

K_{z}(x, y)u^{j}_{0}(y)dy
≤

Z

R^{n}_{+}

1

|x−y|^{n−1}
u^{j}_{0}(y)

dy+ Z

R^{n}_{+}

1

|y−z|^{n−1}
u^{j}_{0}(y)

dy.

By Remark 2.1, Lemma 2.5 and Lemma 2.6, there exists a constant C0 >0 such that

Z

R^{n}+

Kz(x, y)u^{j}_{0}(y)dy
_{L}_{p(x)}_{(}

R^{n}_{+},C`_{n})≤C0ku^{j}_{0}k_{L}p(x)(R^{n}+,C`n). (2.7)

Now let us extendu^{j}_{k}(x) by zero toR^{n}\R^{n}+. Note that the position ofz which is
outside of a half space R^{n}+ leads to the fact that G(y−z) has no singularities for
any y ∈R^{n}+. Thus it is easy to show that _{∂y}^{∂}

kKz(x, y) satisfies the conditions of
Calder´on-Zygmund kernel onR^{n}×R^{n}. In view of Lemma 2.8, there exist positive
constantCk(k= 1, . . . , n) such that

Z

R^{n}+

∂

∂y_{k}K_{z}(x, y)u^{j}_{k}(y)
_{L}_{p(x)}_{(}

R^{n}_{+},C`_{n})≤C_{k}ku^{j}_{k}k_{L}p(x)(R^{n}+,C`n). (2.8)
Combining (2.7) with (2.8), we have

kT u^{j}k_{L}p(x)(R^{n}+,C`n)≤C0ku^{j}_{0}k_{L}p(x)(R^{n}+,C`n)+

n

X

k=1

Ckku^{j}_{k}k_{L}p(x)(R^{n}+,C`n).

Lettingj→ ∞, by means of the Continuous Linear Extension Theorem, the oper-
atorT can be uniquely extended to a bounded linear operatorTesuch that for all
f ∈W^{−1,p(x)}(R^{n}+,C`n), there exists a constantC >e 0 such that

kT fe k_{L}p(x)(R^{n}+,C`_{n})≤C

kf0k_{L}p(x)(R^{n}+,C`_{n})+

n

X

k=1

kfkk_{L}p(x)(R^{n}+,C`_{n})

≤Ckfe k_{W}−1,p(x)(R^{n}+,C`_{n}).

Hence claim (ii) follows.

Lemma 2.10. The following operators are continuous linear operators:

(i) D:W^{1,p(x)}(R^{n}+,C`n)→L^{p(x)}(R^{n}+,C`n).

(ii) De :L^{p(x)}(R^{n}+,C`n)→W^{−1,p(x)}(R^{n}+,C`n).

Proof. (i) The proof is similar to that of [11, Lemma 2.6], so we omit it.

(ii) We consider the following Dirichlet problem of the Poisson equation with homogeneous boundary data

−∆u=f, in R^{n}+

u= 0, on Σ (2.9)

It is easy to see that for allf ∈W^{−1,p(x)}(R^{n}+,C`n) problem (2.9) still has a unique
weak solution u ∈ W^{1,p(x)}(R^{n}+,C`n), see Diening, Lengeler and Ruˇziˇcka [5]. We
denote by ∆^{−1}_{0} the solution operator. On the other hand, it is clear that the
operator

∆ :W^{1,p(x)}(R^{n}+,C`n)→W^{−1,p(x)}(R^{n}+,C`n)

is continuous, so we obtain from Lemma 2.9 that the operator De = −∆T :
L^{p(x)}(R^{n}+,C`n)→W^{−1,p(x)}(R^{n}+,C`n) is continuous, where the operatorDe can be
considered as a unique continuous linear extension of the operatorD.

Lemma 2.11. Let p(x)∈ P(R^{n}+).

(i) Ifu∈W^{1,p(x)}(R^{n}+,C`n), then the Borel-Pompeiu formulaF u(x)+T Du(x) =
u(x)holds for allx∈R^{n}+.

(ii) If u ∈ L^{p(x)}(R^{n}+,C`n), then the equation DT u(x) = u(x) holds for all
x∈R^{n}+.

Proof. Let us denote by C_{0}^{∞}(R^{n}+) the space of all restrictions of functions from
C_{0}^{∞}(R^{n}) toR^{n}+. Furthermore, suppose ϕ∈C_{0}^{∞}(R^{n}+). Now, let us consider a point
y ∈Ω and the open ball B(0, r) with origin 0, radius r, and boundary S(0, r). If
ris sufficiently large such that y lies in the domain Ω(r) =B(0, r)∩R^{n}+. For this
domain, we have

F_{S(0,r)}ϕ(y) =ϕ(y)−T_{Ω(r)}Dϕ(y).

see [25] for more details. This can be written in the form

r→∞lim Z

Σ∩B(0,r)

+ Z

S(0,r)∩R^{n}_{+}

K_{z}(x, y)α(y)u(y)dS_{y}

=ϕ(y)− lim

r→∞T_{Ω(r)}Dϕ(y)
Since

r→∞lim Z

Σ∩B(0,r)

Kz(x, y)α(y)u(y)dSy= Z

Σ

Kz(x, y)α(y)u(y)dSy

and

r→∞lim T_{Ω(r)}Dϕ(y) =T_{R}^{n}

+Dϕ(y), lim

r→∞

Z

S(0,r)∩R^{n}_{+}

K_{z}(x, y)α(y)u(y)dS_{y} = 0,
we obtain the Borel-Pompeiu formula in case ofϕ∈C_{0}^{∞}(R^{n}+). Finally, the desired
result (i) follows immediately from the density document.

(ii) Using the same idea with (i), we can get directly the desired result from [25,

Lemma 2.6].

Lemma 2.12. Let p(x) satisfy (2.1).

(i) If u∈L^{p(x)}(R^{n}+,C`n), thenTeDu(x) =e u(x)for all x∈R^{n}+.
(ii) If u∈W^{−1,p(x)}(R^{n}+,C`n), thenDeT u(x) =e u(x)for allx∈R^{n}+.

Proof. (i) follows from Lemma 2.11 (i) and the denseness of W_{0}^{1,p(x)}(R^{n}+,C`n) in
L^{p(x)}(R^{n}+,C`_{n}).

(ii) follows from Lemma 2.11 (ii) and the denseness ofC_{0}^{∞}(R^{n}+,C`n) in the space
W^{−1,p(x)}(R^{n}+,C`n), see [3, Proposition 12.3.4] for the details.

G¨urlebeck and Spr¨oßig [14, 15] showed that the orthogonal decomposition of the
spaceL^{2}(Ω) holds in the hyper-complex function theory:

L^{2}(Ω,C`_{n}) = (kerD∩L^{2}(Ω,C`_{n}))⊕DW_{0}^{1,2}(Ω,C`_{n}) (2.10)
with respect to the Clifford-valued product (2.3). Note that kerD denotes the set
of all monogenic functions on Ω. This decomposition has a number of applications,
especially to the theory of partial differential equations, see [6] for the complex
case and [14] for the hyper-complex case. K¨ahler [22] extended the orthogonal
decomposition (2.10) to the spacesL^{p}(Ω) in form of a direct decomposition in the
case of Clifford analysis. In [7], Fu et al. extended the direct decomposition to the
case of variable exponent Lebesgue spaces in bounded smooth domains.

Theorem 2.13. The spaceL^{p(x)}(R^{n}+,C`n)allows the Hodge-type decomposition
L^{p(x)}(R^{n}+,C`n) = (kerDe∩L^{p(x)}(R^{n}+,C`n))⊕DW_{0}^{1,p(x)}(R^{n}+,C`n) (2.11)
with respect to the Clifford-valued product (2.3).

Proof. Similar to the proof of [22, Theorem 6], we first show that the intersection
of (kerDe ∩L^{p(x)}(R^{n}+,C`_{n})) and DW_{0}^{1,p(x)}(R^{n}+,C`_{n}) is empty. Suppose f belongs
to both kerDe∩L^{p(x)}(R^{n}+,C`n) andDW_{0}^{1,p(x)}(R^{n}+,C`n), thenDfe = 0 . Becausef
belongs toDW_{0}^{1,p(x)}(R^{n}+,C`n), there exists a function v ∈ W_{0}^{1,p(x)}(R^{n}+,C`n) such
that Dv = f. Hence, we obtain that −∆v = 0 and v = 0 on Σ. From the
uniqueness of ∆^{−1}_{0} we obtainv = 0. Consequently, f = 0. Therefore, the sum of
the two subspaces is a direct one.

Now letu∈ L^{p(x)}(R^{n}+,C`_{n}). Then u_{2} =D∆^{−1}_{0} Due ∈DW_{0}^{1,p(x)}(R^{n}+,C`_{n}). Let
u_{1}=u−u2. Thenu_{1}∈L^{p(x)}(R^{n}+,C`_{n}). Furthermore, we takeu_{k} ∈W_{0}^{1,p(x)}(R^{n}+,C`_{n})
such thatu_{k}→uinL^{p(x)}(R^{n}+,C`_{n}), then by the denseness ofW_{0}^{1,p(x)}(R^{n}+,C`_{n}) in
L^{p(x)}(R^{n}+,C`_{n}) and Lemma 2.2, we have for anyϕ∈W_{0}^{1,p}^{0}^{(x)}(R^{n}+,C`_{n})

u_{1}, Dϕ

C`_{n}= u−u_{2}, Dϕ

C`_{n}

= lim

k→∞ Duk−DD∆^{−1}_{0} Duk, ϕ

C`_{n}

= lim

k→∞ Duk−Duk, ϕ

C`_{n}= 0,

which implies that u1 ∈kerD. Sincee u∈L^{p(x)}(R^{n}+,C`n) is arbitrary, the desired

result follows.

From this decomposition we can get the following two projections
P :L^{p(x)}(R^{n}+,C`_{n})→kerDe∩L^{p(x)}(R^{n}+,C`_{n}),

Q:L^{p(x)}(R^{n}+,C`n)→DW_{0}^{1,p(x)}(R^{n}+,C`n).

Moreover, we have

Q=D∆^{−1}_{0} D, Pe =I−Q.

Corollary 2.14. The spaceL^{p(x)}(R^{n}+,C`_{n})∩imQis a closed subspace ofL^{p(x)}(R^{n}+,C`_{n}).

The proof can be easily done by combining Theorem 2.13, Lemma 2.3 with Lemma 2.10. We refer the reader to [38, Lemma 2.6] for a similar argument.

Corollary 2.15. L^{p(x)}(R^{n}+,C`_{n})∩imQ∗

=L^{p}^{0}^{(x)}(R^{n}+,C`_{n})∩imQ. Namely, the
linear operator

Φ :DW^{1,p}

0(x)

0 (R^{n}+,C`n)→ DW_{0}^{1,p(x)}(R^{n}+,C`n)^{∗}
given by

Φ(Du)(Dϕ) = (Dϕ, Du)Sc:=

Z

R^{n}+

DϕDu

0dx is a Banach space isomorphism.

Proof. In terms of Lemma 2.14, DW_{0}^{1,p(x)}(R^{n}+,C`n) and DW^{1,p}

0(x)

0 (R^{n}+,C`n) are
reflexive Banach spaces since they are closed subspaces in L^{p(x)}(R^{n}+,C`_{n}) and
L^{p}^{0}^{(x)}(R^{n}+,C`n) respectively. The linearity of Φ is clear. For injectivity, suppose

Φ(Du)(Dϕ) = (Dϕ, Du)Sc= 0 (2.12)

for all ϕ ∈ W_{0}^{1,p(x)}(R^{n}+,C`_{n}) and some u ∈ W_{0}^{1,p}^{0}^{(x)}(R^{n}+,C`_{n}). For any ω ∈
L^{p(x)}(R^{n}+,C`_{n}), according to (2.11), we may write ω = α+β with α ∈ kerDe ∩

L^{p(x)}(R^{n}+,C`_{n}) andβ ∈DW_{0}^{1,p(x)}(R^{n}+,C`_{n}). Thus we obtain
(ω, Du)Sc= (α+β, Du)Sc= (β, Du)Sc.

This and (2.12) gives (ω, Du)Sc= 0. This means thatDu= 0. It follows that Φ is
injective. To get subjectivity, letf ∈ DW_{0}^{1,p(x)}(R^{n}+,C`n)∗

. By the Hahn-Banach
Theorem, there isF ∈ L^{p(x)}(R^{n}+,C`_{n})∗

withkFk=kfkandF|_{DW}1,p(x)

0 (R^{n}_{+},C`_{n})=
f. Moreover, there existsϕ∈L^{p}^{0}^{(x)}(R^{n}+,C`_{n}) such thatF(u) = (u, ϕ)_{Sc}for anyu∈
L^{p(x)}(R^{n}+,C`_{n}). According to (2.11), we can writeϕ=η+Dα, whereη∈kerDe∩
L^{p}^{0}^{(x)}(R^{n}+,C`_{n}),Dα∈DW_{0}^{1,p}^{0}^{(x)}(R^{n}+,C`_{n}). For anyDu∈DW_{0}^{1,p(x)}(R^{n}+,C`_{n}), we
have

f(Du) = (Du, ϕ)_{Sc}= (Du, Dα)_{Sc}= Φ(Dα)(Du).

Consequently, Φ(Dα) =f. It follows that Φ is surjective. By [10, Theorem 3.1] we have

|Φ(Du)(Dϕ)| ≤CkDϕk_{L}p(x)(R^{n}+,C`_{n})kDuk_{L}p0(x)(R^{n}+,C`n).

This means that Φ is continuous. Furthermore, it is immediate that Φ^{−1}is contin-
uous from the Inverse Function Theorem. This ends the proof of Lemma 2.3.

3. Stokes equations in the half-space

In the section, we consider the Stokes system which consists in finding a solution (u, π) for

−∆u+ 1

µ∇π= ρ

µf inR^{n}+, (3.1)

divu=f_{0} in R^{n}+, (3.2)

u=v0 on Σ. (3.3)

With R

Ωf0dx = R

∂Ωn·v0dx the necessary condition for the solvability is given.

Here, uis the velocity, π the hydrostatic pressure, ρ the density, µ the viscosity,
f the vector of the external forces and the scalar function f_{0} a measure of the
compressibility of fluid. The boundary condition (3.3) describes the adhesion at
the boundary of the domain Ω for v_{0} = 0. This system describes the stationary
flow of a homogeneous viscous fluid for small Reynold’s numbers. For more details,
we refer to [2, 14, 15, 16, 20].

In this paper, for f =Pn

i=1fiei and u=Pn

i=1uiei, we consider the following Stokes system in the hyper-complex formulation (see [16, 17]):

DDue +1

µDπ= ρ

µf in R^{n}+, (3.4)

[Du]0= 0 in R^{n}+, (3.5)

u= 0 on Σ. (3.6)

Definition 3.1. We say that (u, π) ∈ W_{0}^{1,p(x)}(R^{n}+,C`n)×L^{p(x)}(R^{n}+) is a solu-
tion of (3.4)–(3.6) provided that it satisfies the system (3.4)–(3.6) for all f ∈
W^{−1,p(x)}(R^{n}+,C`_{n}).

Definition 3.2. The operator∇e :L^{p(x)}(R^{n}+)→(W^{−1,p(x)}(R^{n}+))^{n} is defined by
h∇f, ϕie =−hf,divϕi:=−

Z

R^{n}+

fdivϕdx
for allf ∈L^{p(x)}(R^{n}+) andϕ∈(C_{0}^{∞}(R^{n}+))^{n}.

Theorem 3.3. Suppose f ∈ W^{−1,p(x)}(R^{n}+,C`n). Then the Stokes system (3.4)–

(3.6)has a unique solution(u, π)∈W_{0}^{1,p(x)}(R^{n}+,C`n)×L^{p(x)}(R^{n}+) of the form
u+ 1

µT Qπ= ρ µT QT f,e with respect to the estimate

kDuk_{L}p(x)(R^{n}+,C`n)+ 1

µkQπk_{L}p(x)(R^{n}+)≤Cρ

µkQT fe k_{L}p(x)(R^{n}+,C`n).

Here,C≥1is a constant and the hydrostatic pressureπis unique up to a constant.

Proof. We first prove that iff ∈W^{−1,p(x)}(R^{n}+,C`_{n}), then we have the representa-
tion

T QT fe =u+T Qω.

Indeed, let ϕn ∈ W_{0}^{1,p(x)}(R^{n}+,C`n) with ϕn → ϕ in L^{p(x)}(R^{n}+,C`n). By Lemma
2.11, we have

T QT(Dϕ_{n}) =T Qϕ_{n}.

SinceW_{0}^{1,p(x)}(R^{n}+,C`n) is dense inL^{p(x)}(R^{n}+,C`n), it follows thatT QTeDϕe =T Qϕ.

Thus, foru∈W_{0}^{1,p(x)}(R^{n}+,C`n) andπ∈L^{p(x)}(R^{n}+) we obtain
T QTe(ρ

µf) =T QTe(DDue + 1

µDπ) =e u+ 1 µT Qπ.

This implies that our system (3.4)–(3.5) is equivalent to the system u+ 1

µQT Qπ= ρ

µT QT f,e (3.7)

[Qπ]0= [QT f]e 0. (3.8)

Obviously, the equality (3.4) is equivalent to the equality Du+ 1

µQπ= ρ

µQT f.e (3.9)

Now we need to show that for each f ∈ W^{−1,p(x)}(R^{n}+,C`^{1}_{n}), the function QT f
can be decomposed into two functions Du and Qπ. Suppose Du+Qπ = 0 for
u ∈ W_{0}^{1,p(x)}(R^{n}+,C`^{1}_{n})∩ker div and π ∈ L^{p(x)}(R^{n}+). Then (3.5) gives [Qπ]0 = 0.

Thus,Qπ = 0. Hence, Du=Qπ = 0. This means thatDu+Qπ is a direct sum, which is a subset of imQ.

Next we have to ask about the existence of a functionalF ∈(L^{p(x)}(R^{n}+,C`^{1}_{n})∩
imQ)^{∗} with F(Du) = 0 andF(Qπ) = 0 but F(QT f)e 6= 0. This is equivalent to
ask if there existsg∈W^{−1,p}^{0}^{(x)}(R^{n}+,C`^{1}_{n}), such that for allu∈W_{0}^{1,p(x)}(R^{n}+,C`^{1}_{n})∩
ker div andω∈L^{p(x)}(R^{n}+),

(Du, QT g)e Sc= 0, (3.10)

(Qπ, QT g)e _{Sc}= 0, (3.11)

but (QT f, Qe T g)e Sc6= 0. Here, Lemmas 2.9 and Corollary 2.15 are employed.

Thus, let us consider the system (3.10) and (3.11) with g ∈ W^{−1,p}^{0}^{(x)}(Ω,C`^{1}_{n})
for all open cubes Ω⊂R^{n}+. Notice that, with the help of Lemma 2.10, (3.10) yields

(Du, QT g)e Sc = (u,DQe T g)e Sc= (u,DeT ge −DPe T g)e Sc = (u, g)Sc= 0,
which implies g = ∇he = Dhe with h ∈ L^{p}_{loc}^{0}^{(x)}(R^{n}+) because of [38, Lemma 2.8].

Furthermore, by Definition 3.2, it is easy to see that if g ∈ W^{−1,p}^{0}^{(x)}(R^{n}+,C`^{1}_{n}),
thenh∈L^{p}^{0}^{(x)}(R^{n}+). Thus it follows from (3.11) and Lemma 2.3,

(Qπ, QT g)e _{Sc}= (Qπ, QTeDh)e _{Sc}= (Qπ, Qh)_{Sc}= 0

holds for eachπ∈L^{p(x)}(R^{n}+). Hence, Qπ=|Qh|^{p}^{0}^{(x)−2}QhgivesQh= 0. Then we
obtain

g=Dhe =DQhe +DP he = 0.

Furthermore, we obtain

(QT f, Qe T g)e Sc= 0, for allf ∈W^{−1,p(x)}(R^{n}+,C`^{1}_{n}).

Finally, (3.9) yields

kDuk_{L}p(x)(R^{n}+,C`n)+ 1

µkQπk_{L}p(x)(R^{n}+)≥ ρ

µkQT fe k_{L}p(x)(R^{n}+,C`n).
By the Norm Equivalence Theorem, we obtain

kDuk_{L}p(x)(R^{n}+,C`n)+ 1

µkQπk_{L}p(x)(R^{n}+)≤Cρ

µkQT fe k_{L}p(x)(R^{n}+,C`n).
By Remark 2.1, Lemma 2.9 and the boundedness of the operatorQ, we obtain

kuk_{W}1,p(x)

0 (R^{n}_{+},C`_{n})+ 1

µkQπk_{L}p(x)(R^{n}+)≤Cρ

µkfk_{W}−1,p(x)(R^{n}+,C`n), (3.12)
which implies the uniqueness of solution. Note that Qπ = 0 implies π ∈ kerD.e
Therefore,πis unique up to a constant. The proof is complete.

4. N-S equations in the half-space

In this section, we consider the time-independent Navier-Stokes equations in variable exponent spaces of Clifford-valued functions in a half-space:

−∆u+ρ

µ(u· ∇)u+1

µ∇π= ρ

µf inR^{n}+, (4.1)

divu=f0 in R^{n}+, (4.2)

u=v0 on Σ. (4.3)

In addition to the case of the Stokes system, the main difference from the above-
mentioned Stokes equations is the appearance of the non-linear convection term
(u· ∇)u. In 1928, Oseen showed that one can get relatively good results if the
convection term (u· ∇)u is replaced by (v · ∇)u , where v is a solution of the
corresponding Stokes equations. In 1965, Finn [13] proved the existence of solutions
for small external forces with a spatial decreasing to infinity of order|x|^{−1} for the
case ofn= 3, and used the Banach fixed-pointed theorem. G¨urlebeck and Spr¨oßig
[14, 15, 17] solved this system by a reduction to a sequence of Stokes problems
provided the external force f belongs to L^{p}(Ω,H) for a bounded domain Ω and
6/5 < p < 3/2. Cerejeiras and K¨ahler [2] obtained the similar results provided